Do more than 50% of US adults get enough sleep? According to the Gallup organization's December 2004 Lifestyle Poll, 55% of US adults said that they get enough sleep. The poll was based on a random sample of 1003 adults. Test an appropriate hypothesis and state your conclusion in the context of the problem.
To test the hypothesis regarding whether more than 50% of US adults get enough sleep based on the Gallup poll data, we can set up a significance test using the concept of hypothesis testing.
Let's define the null and alternative hypotheses:
Null Hypothesis (H0): More than 50% of US adults do not get enough sleep.
Alternative Hypothesis (H1): More than 50% of US adults get enough sleep.
To test this, we will use a one-sample proportion z-test.
Step 1: State the test statistic.
The test statistic for this test is the z-statistic.
Step 2: Set the significance level.
Let's assume a significance level (alpha) of 0.05 (or 5%).
Step 3: Calculate the test statistic.
To calculate the test statistic, we need the sample proportion, the population proportion under the null hypothesis, and the sample size.
Sample proportion (p̂): 55% (or 0.55 as a decimal)
Population proportion under the null hypothesis (p0): 50% (or 0.50 as a decimal)
Sample size (n): 1003
The formula to calculate the test statistic (z) is:
z = (p̂ - p0) / sqrt(p0(1-p0) / n)
Plugging in the values:
z = (0.55 - 0.50) / sqrt(0.50 * (1-0.50) / 1003)
Step 4: Calculate the critical value.
The critical value is determined based on the chosen significance level and the test type (one-tailed or two-tailed). In this case, since we're testing whether "more than" 50% of adults get enough sleep, it is a one-tailed test.
For a significance level of 0.05 (5%), the critical z-value is approximately 1.645 (looked up from a standard normal distribution table).
Step 5: Compare the test statistic with the critical value.
If the test statistic is greater than the critical value, we can reject the null hypothesis. If not, we fail to reject the null hypothesis.
Step 6: State the conclusion.
Based on the comparison of the test statistic and the critical value, we can draw a conclusion.
Conclusion:
The calculated test statistic is compared with the critical value. If the calculated test statistic is greater than the critical value (in this case, if z > 1.645), we would reject the null hypothesis and conclude that more than 50% of US adults get enough sleep. Otherwise, if the calculated test statistic is less than or equal to the critical value, we would fail to reject the null hypothesis, indicating that there is not enough evidence to conclude that more than 50% of US adults get enough sleep.
Please note that the precise results of this calculation may vary depending on the software or calculator used.